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I read the article of Steven Hair "A degree-theoretic proof of a coarse fixed point principle". I have the following question.

I start with some main definitions from this article. A coarse map $X \to Y$ is a function (where $X$ and $Y$ are metric spaces) such that

  1. If $B\subset Y$ is bounded, $f^{-1}(B) \subset X$ is bounded ($f$ is a proper)

  2. For every $R>0$ there exists $S>0$, such that for $x, x' \in X$ with $d_X(x,x') \leq R$, then $d_Y(f(x),f(x')) \leq S$ ($f$ is bornologous map).

Let X be a compact metrizable space. The open cone is denoted as $\mathcal{O}X$ (space obtained from $[0,1) \times X$ by collapsing $\left\{0\right\} \times X$ to a point).

There are following definitions of Strong coarse fixed point property and Weak coarse fixed point property:

  1. A coarse map $f:\mathcal{O}X \to \mathcal{O}X$ has the strong coarse fixed point property (SCFPP) if there exists a point $\xi \in \mathcal{O}X$ (called the coarse fixed point of $f$), a constant $R>0$, and a sequence of points $x_i \to \infty$ in $\mathcal{O}X$ such that, for all $i$, both $x_i$ and $f_i$ are within distance $R$ of the ray $\mathcal{O}\xi$

  2. A coarse map $f:\mathcal{O}X \to \mathcal{O}X$ has the Weak coarse fixed point property (WCFPP) if there exists $\xi \in X$ such that, for any neighborhood $U$ of $\xi$, there is a sequence of points $x_i \to \infty$ in $\mathcal{O}X$ such that, for all $i$, $x_i$ and $f(x_i)$ lie in the cone $\mathcal{O}U$

I'm trying to find an example which has the Weak coarse fixed point property (WCFPP) but hasn't the strong coarse fixed point property (SCFPP). Can you help me please? Thank you

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